On Jutila's integral in the circle problem

M. A. Korolev ; D. A. Popov

Izvestiya. Mathematics , Tome 86 (2022) no. 3, pp. 413-455

Résumé

We study a ‘correlation’ function $\mathcal{K}_{P} = \mathcal{K}_{P}(T;H,U)$ of the error term $P(t)$ in the circle problem, that is, the integral of the product $P(t)P(t+U)$ over the interval $(T,T+H]$, $1\,{\le}\, U, H\,{\le}\, T$. The case of small $U$, $1\le U\ll \sqrt{T}$, was in essence studied by Jutila in 1984. It turns out that, for all these $U$ and sufficiently large $H$, $\mathcal{K}_{P}$ attains its maximum possible value. In this paper we study the case of ‘large’ $U$, $\sqrt{T}\ll U\le T$, when the behaviour of $\mathcal{K}_{P}$ becomes more complicated. In particular, we prove that the correlation function may be positive and negative of maximally large modulus as well as having very small modulus on sets of values of $U$ of positive measure.

Détail

Keywords: circle problem, correlation function, simultaneous approximations.
Mots-clés : Jutila's conjecture, Jutila's formula